Let $W(\alpha)$ denote the set of all (countable) ordinals *writable* by Ordinal Turing Machines with a single (countable) parameter $\alpha$, i.e. each computation starts with a single ($\alpha$-th) cell on the input tape marked with a non-zero symbol (all other cells are marked with a zero symbol).

Question: does there exist an ordinal $\beta$ such that there exists at least one ordinal $\gamma < \beta$ such that $\gamma \notin W(\beta)?$ If no, why? If yes, how large is the smallest such $\beta$?